13. Construct a square with side 2 in., and in it inscribe a circle. Join points of contact, and show by tests that the resulting figure is a square. What is its side? 14. Construct a rhombus with sides 50 millimetres in length and angles 75° and 105°, and describe a circle touching the sides. 15. Construct a rhombus with diagonals of 60 and 80 millimetres, and in it inscribe a circle. Measure length of radius, and test accuracy of measurement by calculation. 16. Construct a square with side of 2 in., and about it describe a circle. At the angular points of the square draw tangents to the circle, and by tests show that the resulting figure is a square. 17. Construct a rectangle with sides 30 and 40 millimetres, and about it describe a circle. Measure radius of circle, and test accuracy of measurement by calculation. 18. Construct a rectangle such that when a circle is described about it, and tangents drawn at the angular points, the resulting rhombus shall have angles of 60° and 120°. 19. Beginning with a circle of radius 50 millimetres, inscribe a square in it, then a circle within the square, and finally a square within this latter circle. Test the accuracy of the final square. What are the lengths of the sides of the squares, and the length of the radius of the second circle? 20. About a circle of radius 12 in. describe a quadrilateral with angles 60°, 150°, 110°, 40°. Can you describe about a circle a quadrilateral equiangular to any given quadrilateral ? CHAPTER XVII. Regular Polygons. 1. A polygon is a rectilineal figure contained by more than four straight sides. A pentagon is a figure of 5 sides. A polygon is said to be regular when all its sides are equal, and also its angles equal. 2. The angles at any point, for example, at the centre of a circle, make up 360°. We can divide this interval, by means of the protractor, into a number, 5, 6, 8, of equal angles. If we prolong the sides of these angles until they intersect the circumference of the circle, and join the successive points of intersection, we have a regular polygon of 5, 6, 8, . . . .... 9 sides, as the case may be. E 3. To describe a regular pentagon in a circle: A pentagon having five sides, the angle subtended at the centre of the circle by the side of a regular pentagon inscribed in the circle, will be of 360° 72°. Using then the protractor, or adjusting the bevel to an angle of 72°, lay off at the centre 5 angles, each of this magnitude. = 72° Produce the sides of the angles to meet the circumference, and join the succeeding points of intersection. The construction being accurately made, the bevel will show the equality of the angles ABC, BCD, and the dividers will show the equality of the sides AB, BC. Of course, the evident equality of the isosceles triangles OAB, OBC, . . ., , proves the equality of the sides and angles of the pentagon. The angle at the vertex of each isosceles triangle in the figure being 72°, each angle at the base must be 54°; and therefore each of the angles (ABC, BCD, . . . ) of a regular pentagon is 108°. K 4. If tangents to the circle be drawn at the angular points of the pentagon ABCDE, the tangents form another regular pentagon, which is said to be about the circle. The equality of the sides FG, GH, .. may be tested with the dividers, and the equality of the angles FGH, GHK, Iwith the bevel. ... D H D 72° G B եւ 5. If we wish to construct on a given straight line (AB), as side, a regular pentagon, at the points A and B, with the protractor we mark off angles BAE, ABC of 108°, and with the dividers make BC and AE, each equal to AB. At C we again make an angle BCD of 108°, and mark off CD equal to AB. 108 108 108° Joining E and D, we have a regular pentagon ABCDE. Using the bevel, we shall find that the angles at E and D are equal to the three other angles, and the dividers will prove the side DE to be equal to the other sides. The radius of a circle being 36 millimetres, inscribe in it a regular pentagon. With the dividers and bevel prove the accuracy of your construction, that the sides and angles are equal. Describe also about the same circle a regular pentagon. With the dividers and bevel prove the accuracy of your construction. On a line of length 2 inches, as side, construct a regular pentagon. With instruments prove the accuracy of your construction. Exercises. 1. In a circle of radius 32 millimetres, inscribe a regular pentagon. Test equality of sides with dividers, and equality of angles with bevel or protractor. 2. In a circle of radius 12 in., inscribe a regular pentagon. Test accuracy of construction. 3. About a circle of radius 1 in., describe a regular pentagon. Test accuracy of construction. 4. About a circle of radius & in., describe a regular pentagon. Test accuracy of construction. 5. In the two preceding questions, where the radius of one circle is twice that of the other, examine the relation between the lengths of all corresponding lines that can be drawn in the two figures,-sides, lines joining non-adjacent angles, segments of these lines by their intersection. 6. Inscribe two regular pentagons in any two circles of different radii. With the bevel examine the relation between all corresponding angles that can be formed in the two figures. 7. Describe an irregular equilateral pentagon, each side being 1 in. 8. About a circle of radius 11⁄2 in., describe a pentagon with angles 80°, 110°, 145°, 70°, and 135°. 9. Describe a regular pentagon with side of 1 in. of construction. 10. Describe a regular pentagon with side of 2 in. of construction. Test accuracy Test accuracy 11. In the two preceding questions, what is the relation between the radii of the two circles about the pentagons? 12. Hence if you have in a circle (radius OA) a regular pentagon with side 30 millimetres, how many times OA should you make the radius of a second circle, that the side of a regular pentagon in it may be 45 millimetres? 13. ABCDE being a regular pentagon, what sort of triangles are ACD, and ABC? What are the magnitudes of the angles CAD, ACD, CBD? 14. In the figure of the preceding question, join each angle to the other angles. Is the pentagon thus obtained, in the centre of the Measure each angle of the figure, and assign to it its magnitude in figure, regular? Apply tests. formed by intersecting lines, degrees. 15. Since the side of a regular pentagon subtends an angle of 72° at the centre of the circle about it, what angle should a side subtend at the circumference? Hence assign to each angle at circumference in question 14, its proper magnitude, and deduce values of all other angles in the figure. 16. In the figure of question 14, indicate all lines that are equal to one another; also all triangles that are isosceles. 17. In the same figure erase the circumference, and sides of the pentagon, so obtaining a star-shaped figure. Show how such a figure (called a pentagram) could be described without taking the pencil from the paper. 18. Without describing a circle, construct a pentagram, the line corresponding to AC being 3 in. Test accuracy of construction by determining lengths AB, BC, and angles ABC, BCD, 19. In the figure of question 14, how many rhombuses are there? 20. With respect to how many lines is a regular pentagon symmetrical? Has it central symmetry? |